Rees matrix semigroup
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In mathematics, the Rees matrix semigroups are a special class of semigroups introduced by David Rees in 1940. They are of fundamental importance in semigroup theory because they are used to classify certain classes of simple semigroups.
Definition
[edit]Let S be a semigroup, I and Λ non-empty sets and P a matrix indexed by I and Λ with entries pλ,i taken from S. Then the Rees matrix semigroup M(S; I, Λ; P) is the set I×S×Λ together with the multiplication
- (i, s, λ)(j, t, μ) = (i, s pλ,j t, μ).
Rees matrix semigroups are an important technique for building new semigroups out of old ones.
Rees' theorem
[edit]In his 1940 paper Rees proved the following theorem characterising completely simple semigroups:
A semigroup is completely simple if and only if it is isomorphic to a Rees matrix semigroup over a group.
That is, every completely simple semigroup is isomorphic to a semigroup of the form M(G; I, Λ; P) for some group G. Moreover, Rees proved that if G is a group and G0 is the semigroup obtained from G by attaching a zero element, then M(G0; I, Λ; P) is a regular semigroup if and only if every row and column of the matrix P contains an element that is not 0. If such an M(G0; I, Λ; P) is regular, then it is also completely 0-simple.
See also
[edit]References
[edit]- Rees, David (1940), On semi-groups, vol. 3, Proc. Camb. Philos. Soc., pp. 387–400.
- Howie, John M. (1995), Fundamentals of Semigroup Theory, Clarendon Press, ISBN 0-19-851194-9.